TRANSACTIONS of the AMERICAN MATHEMATICAL SOCIETY Volume 240, lune 1978
ERICSSON'SCONJECTURE ON THE RATEOF ESCAPEOF ¿-DIMENSIONALRANDOM WALK BY HARRYKESTEN
Abstract. We prove a strengthened form of a conjecture of Erickson to the effect that any genuinely ¿-dimensional random walk S„, d > 3, goes to infinity at least as fast as a simple random walk or Brownian motion in dimension d. More precisely, if 5* is a simple random walk and B, a Brownian motion in dimension d, and \j/i [l,oo)-»(0,oo) a function for which /"'/^(OiO, then \¡/(n)~'\S*\-» oo w.p.l, or equivalent^, ifr(t)~'\Bt\ -»oo w.p.l, iff ffif/(tY~2t~d/2 < oo; if this is the case, then also $(ri)~'\S„\ -» oo w.p.l for any random walk Sn of dimension d.
1. Introduction. Let XX,X2,... be independent identically distributed ran- dom ¿-dimensional vectors and S,, = 2"= XX¡.Assume throughout that d > 3 and that the distribution function F of Xx satisfies supp(P) is not contained in any hyperplane. (1.1) The celebrated Chung-Fuchs recurrence criterion [1, Theorem 6] implies that1 |5n|->oo w.p.l irrespective of F. In [4] Erickson made the much stronger conjecture that there should even exist a uniform escape rate for S„, viz. that «""iSj -* oo w.p.l for all a < \ and all F satisfying (1.1). Erickson proved his conjecture in special cases and proved in all cases that «-a|5„| -» oo w.p.l for a < 1/2 — l/d. Our principal result is the following theorem which contains Erickson's conjecture and which makes precise the intuitive idea that a simple random walk S* (which corresponds to the distribution F* which puts mass (2d)~x at each of the points (0, 0,..., ± 1, 0,..., 0)) goes to infinity slower than any other random walk. Theorem. Let d > 3, S„ and S* as above, and let [B,)t>0 be a d-dimen- sional Brownian motion (EB, = 0, EB,(i)Bt(j) = t8¡f). Assume that the func- tion \[i: [ l,oo) -» (0,oo) satisfies rl/\(t)iO. (1.2) Then \¡/(n)~x\S*\-» oo w.p.l and\p(t)~x\B,\ -> oo w.p.l are both equivalent to
Receivedby the editors June 25, 1976. AMS (MOS) subjectclassifications (1970). Primary 60J15, 60F20. Key wordsand phrases. Random walk, escape rate, concentration functions. 'For any vector c e R2(/)}l/2."w.p.l" stands for "with probability one". O American Mathematical Society 1978
65
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 66 HARRY KESTEN f^(t)d-2rd/2< oo. (1-3)
If (1.2) and (1.3) hold, then also ^(/z)-1!^! -» oo w.p.l whenever F satisfies (1.1). As we shall see this theorem is almost immediate from Proposition 1. If d > 3 and F satisfies (1.1) then there exist constants 0 < r,(P), T2(P) < oo such that for all k > 1 and all A > 0, P {|Sn| < A for some 2k < n < 2k+x)
< TX(F){((A + l)2-k/2)"-2 + exp -T2(F)k}. (1.4) It is worthwhile contrasting the theorem with the observation (see [4, end of §5]2)that even if (1.1)—(1.3)hold, there may exist a deterministic sequence of vectors an such that hminf^(n)-x\Sn-an\=0 w.p.l. (1.5) Thus, the theorem is not merely a matter of estimating concentration func- tions for S„. Nevertheless the result is intimately connected with concentra- tion functions. Erickson's proof for \p(t) = t", a < 1/2 - \/d, uses a con- centration function inequality of Esseen [6], and our proof makes heavy use of the following inequalities for concentration functions in dimension two. These estimates too are closely related to those of Esseen [5, Theorem 3], but are more generally applicable. We note that Corollary 1 shows that in the identically distributed case the concentration function decreases like n~\ Proposition 2. Let Z„ Z2,... ,Z„ be independent random two-vectors with distribution functions Gx, G2,..., G„. Let p, p„ p2,..., p„ be strictly positive numbers such that p, < p, and let AX,A2,... ,A„ be sets in R2 X R2 such that A¡ C {« G R2: |w| > p,}2.Define the symmetrizationGf of G¡ by* G*(B)=fGi(B + u)dGi(u) (1.6) and pur
q¡ = ¡ dGf(u), a2= inf / (0,u)2dG*(u), J\u\>o, 1*1-1•>!<&
C=ï{o2 + £[ dG°(u)dG?(v)\. (1.7) ,-1 I ?, Ju,t>eA, J
2More details are given in the recent article by K. B. Erickson, Slowing down d-dimensional randomwalks, Ann. Probability5 (1977),645-651. 3For a distribution function F and Borel set A, F (A) denotes the mass assigned to A by the Borel measure induced by F. 4<«,o> denotes the usual inner product of two vectors of the same dimension.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ¿-DIMENSIONALRANDOM WALK 67
Finally, denote the angle between two vectors u and v, determined in such a way that it lies in [0,tt], by sup P 2z,+ Kxp2 < exp (¿¿-i dGl(u)dG¡(v) \¡=\ Lq¡ Ju.vSA, •log 1 + (1.8) \ \v\ \sin Corollary 1. Let Z„ ... ,Z„ be independent random two-vectors, all with the same distribution function G, and define Gs as in (1.6). If p > 0 and q = f dGs (u), a2 = inf f <0,«>2dGs (u), and if there exist sets BX,B2c [z E R2: \z\ > p) and a constant c > 0 such that \v\ ¡sin sup P 2z,+ <— p2(l+pc-x)L2 + £ ( dGs(u) [ dGs(v)\ .(1.10) n [ q JBi jb2 ) Acknowledgement. We wish to thank Professor Erickson for making his results [4] available to us before publication and for some helpful conversa- tions. 2. Reduction to a two dimensional problem. In this section we shall reduce the proof of Proposition 1 to a two dimensional problem. This problem will then be settled together with Proposition 2, by means of estimates on two dimensional concentration functions in §3. Throughout Kx, K2,... will be strictly positive constants which depend on the dimension d only and whose numerical value is immaterial for our purposes. They will not necessarily have the same value at each appearance. r„ T2,... and kx,1> ^2>k will be con- stants which depend on F and d or some other parameters; where necessary these parameters will be indicated explicitly, except that we shall not indicate dependence on d. For any random variable Y we write Ys = Y — Y' where License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 68 HARRYKESTEN Y' is independent of Y and has the same distribution as Y. If Y has distribution function G, then Ys has the distribution function Gs given by G'(B)=fG(B+y)dG(y). The first reduction shows that we may assume that F has certain smooth- ness properties. It is of a purely technical nature and has little to do with the basic idea of the proof of Proposition 1. The reader should skip the proof of Lemma 1 at first reading. Lemma 1. It suffices to prove Proposition 1 in the case where F(dx) > adx on C0 (2.1) for some a > 0 and some closed cube C0 C Rd (which does not reduce to a point). Proof. Let F satisfy (1.1). We shall construct an P, which satisfies (2.1) (and a fortiori (1.1)) such that Proposition 1 holds for the original S„ as soon as it holds for a random walk, S„(FX)say, whose increments have distribution function P,. For this purpose we first find a large cube C, = [z E Rd:-L < z(i) < L,l< i < d) (2.2) such that F(CX) > 0 and such that the intersection of supp(P) with C, is not contained in any hyperplane. This is possible by (1.1). Define the distribution functions G and H by5 Po = ÏF(Ci), G (S) = (2p0)-xF(S n C,) and ,,, , H(S) = (i -/><,)-'{ÎW n c,) + F(s n C{)}. Then we can write F = p0G + (1 — p0)H, and if IX,I2,..., UX,U2,..., VX,V2,... are totally independent random variables with P{lj = 0) = l-P{lj=l}=Po, P{UjES} = G(S), P{VjES) = H(S), then the joint distribution of {X„)„>x is the same as of {U„(l — I„) + V„I„)„>X(compare [9, p. 1184]).We shall therefore assume that U,V and / are defined on our original probability space and that X„ = Un(\ — I„) + VnIn. Let o, < a2 < ... be the successive (random) indices for which /„ = 1. Then, with o0 = 0, the oi+x — o, are independent and all with the distribution P{oi+x - a,. = r) =Por"'(l -Pol r > 1. (2.3) 5For any set S c R^, Sc denotes its complement, i.e., Rrf\ S. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ¿/-DIMENSIONALRANDOM WALK 69 Even more, if % is the o-field generated by { Uj, V},I}: 1 < j < n), then for each {%) stopping time Tand o*(T) = smallest o¡ which exceeds T, one has P{o*(T) = T+ r\5T) =Por-,0 "Po). r > 1. It is also not hard to see that the {Sc+i — So}i>0 are independent and all with the same distribution function6 F2= 2 P{°x = r)G*-» * H= 1 ^-1(1 -Po)G^-~) * H, r-1 r-1 and that also, conditional on fr, Sa,fT)- ST has the distribution F2 on the set {T< oo}. Now take T = ini{n E[2k,2k+X) mth\Sn\< A) (= oo if no such T exists). Then, since |5^r| < A, P{\S0¡\< 2A for some/ e[I(l -/>0)2*,2(1 - Po)2k+2)} > P{T Now for some yl0 = ^r/^") < °° an^ a31A > A& F2({z ERd: \z\ /.=[i(l-/>o)2*], /2-[2(l-Po^+2]. the last term in (2.4) is at most P{a,t > 2* or a,2 < 2k+2} < T3 exp -T42*, for some r3,r4 < oo depending onp0 only (by (2.3) and standard exponential 6G * H denotes the convolution of G and H, and C*(I) denotes the i-fold convolution of G with itself. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 70 HARRYKESTEN estimates; compare [9, formulae (5.40) - (5.42)]). Thus (2.4) yields for A > Aq, P{\S„\ < A for some 2* < n < 2*+1} = P { T < oo} < 2P {\S„\ < 2A for some / G [ \ (1 - />„)2*,2(1- p0)2k+2)} +2plk + 2T3exp - T42*. (2.5) It is clear from (2.5) that if Proposition 1 holds for the random walk S„(Fj) = S„, whose increments SL — Sa¡ all have the distribution F2, then Proposition 1 also holds for the original S„. Actually (2.5) was derived only for A > A0. However, for A < A0 or even A < 2k/s, (1.4) is immediate from Esseen's estimate [6, Corollary to Theorem 6.2] P{\S„\ < T5(F)(A + \)dn-á'\ (2.6) for some T5(F) < oo. F2 itself does not have to satisfy (2.1). However, set G2-%Prl(l-Po)G«'-l\ (2-7) r=l so that F2 = G2* H, and assume that we can find a distribution function 07, with a continuous density such that f x" dGx(x) = [ x" dG2(x) for ||i>||< 16. (2.8) (Here we use the standard multi-index notation; x" = Ud.xx(i)Hi),for posi- tive integers v(i) and ||i-|| = 2f,,K0-) ^e c^m triat x^ienF\ = Gx * H has the required properties. Before constructing Gx we shall prove this claim. (2.1) is obvious for P,; indeed Gx has a continuous density and hence P, has a density which is lower semicontinuous. Thus we merely have to prove that the validity of Proposition 1 for Sn(Fx) implies the validity of Proposition 1 for S„(F2) (since we already showed that Proposition 1 then also holds for our original S„). Now fix k and A > 2k/s; we already saw above that (1.4) follows from (2.6) if A < 2k/i so that these are the only values of interest. Let #, Xu 30»• • ■ De independent random ¿/-vectors, also independent of {Sai)l>0, and such that N has a normal distribution with mean zero and covariance matrix k~xA2times the identity matrix, and such that each x¡ has distribution P,. Then, for any n, License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ¿/-DIMENSIONALRANDOM WALK 71 P{|S0J< 2Á] < P{|50ii0)|< 2^,1 < j < d] < P{\San(j)+ iV0)|< 3^,1 A2 < P{\S„m(j)+ N(f)\<3A,\ Similarly 2 *(/)+ *(/)< 3,4,1< f < d 1-1 < 4dx'2A + K3 exp - ^ < p 2xi (2.10) If exp(-£|0|2) Thus, by the inversion formula, p{|S^C/)+ #C/)|<3¿,l - P <3A,l < j < d /-i2xC/) + *C/) ¿ I sin 3(L4 0,• • •#„ n ' °j J-'\ •| < ( t )vr • • • /_>'( But, by virtue of (2.8), License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 72 HARRY RESTEN M*) - 45 r(e,xy ,«0,x> d(Gx(x) - G2(x)) r=0 ±f<0,x)X6d(Gx(x) + G2(x))= -^f<0,x}X6dG2(x) <-ih\9\2 1/1,16 ^Po-l^-Po)((r-W 16 16! r-i (see (2.7) and recall that G is concentrated on C,). Consequently, for A > 2*/8 and n < 2k+2, P{\S0ii(j) + N(j)\<3A,Kj -P 2 xU) + ^CO < 3¿, i< / < ¿ i-l r6(P)^"/ • • • J>|,6exp(- £ |0|j ¿0,. • • 4, < T1(F)nAd-d-X6k^d+X6^2< r7(P)^C+16)/22-fc+2. Combinedwith (2.9)and (2.10)this yields < 4dx'2A + T¿F)e-k'\ (2.11) P{\S0j<2A} Interchanging the subscripts 1 and 2 we prove similarly P{\Saj for all A > 2*/8, n < 2k+2.Finally, to prove our claim we appeal to the proof of Theorem3 in [9].Just as in the estimateson pp. 1179-1181of [9], P{\Sa} < A for some 2k < n < 2*+1} < 2P{ftJ<¿ 2 p{\Sa}<2A), (2.13) \n = 0 )'2*+2_,n-2" and also License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ¿/-DIMENSIONAL RANDOM WALK 73 2x < 4dx'2A for some 2k < n < 2*+2 2* + 2 -' 2*+2-l2 p 2^ 2x < %dx'2Aw 2x, < 4dx'2A n=0 1 J « = 2* 2* 12*+2_, 2 p > K4 2^ 2x <{d~x/2A 2x <4dx'2A . (2.14) n-0 i n-2k i Since 2- 2 P{|Sj<,4}->oo as/c->ooand,4 >2*/8, n=0 it followsfrom (2.11)-(2.14)that for /c > rc,(P), P {|50J < A for some 2* < » < 2*+1} -i < 4dx/2A for some 2* < n < 2k+x <2K4 2xi + P < 4dx'2A for some 2*+1 < « < 2*+2 2x,i + r9(F)exp(-rc/2). Since 2"x, can be taken for S„(FX)(its increments Xnhave distribution Fx), we see from this that if Proposition 1 holds for Sn(Fx), then it also holds for S„(F2) and for the original S„. (Note that we can always obtain (1.4) for k < kx by increasing T,.) To complete the proof of the lemma we show finally that there exists a Gx with a continuous density and satisfying (2.8). Let there be M positive integer ¿/-vectors v with 1 < \\p\\ < 16 and consider the set 9fc = {(*,). distributionfunction P on Rrfand all 1 <||i»||< 16J, where the constants A„ < oo are chosen large enough such that p^^fx'dG^x) G[-AA], 1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 74 HARRY RESTEN Assume first that p E 3 9H. Then there exists a supporting hyperplane of 911 at p (see [3, Theorem 8]), i.e., there exist constants c0,c„, 1 < ||i>|| < 16, not all zero, such that 2 cvzv+ c0 > 0 for all z G <3H, (2.15) 1<|M|<16 and 2 cfp„ + c0 = 0. (2.16) l<||,||<16 Then, if M P(x) = 2 c„*'+ c0. ,= 1 (2.15) implies that fP(x)dR(x) > 0 for all distribution functions R on Rrf with 1/x"dR(x) S cp(kx + •■■ + kd+xyX6(kxyx + •■■+ kd+xyd+xy= 0. 1<|H|<16 If we let k¡ -» oo such that d+i k¡(kx+ • • • + kd+x) '-*\ > 0 with 2 \= h we obtain 2 cy(Xxyx+ ■•■ +Xd+xyd+xy=0 H-16 for all A,> 0 with2fí,'A, = 1,i.e., 2 c„x"= 0 H =16 for all x in the closed convex hull of yx,... ,yd+x. Since these (d + 1) points do not lie in any hyperplane, their convex hull has a nonempty interior (see [3, Theorem 4]) and we conclude c„ = 0 for \\v\\ «■ 16. But then also License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ¿/-DIMENSIONALRANDOM WALK 75 2 cf(kx + ■■■ + kd+xyl5(kxyx + ■■■ + kd+xyd+x)'= 0, 1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 76 HARRY KESTEN then t(u,k) P{\(Sn,o>)\<2Ak} > P{\(Sn_x,«y\ whenever k > k2(F), n < T(k) and w = wy- with 2 < / < d (2.25) for a suitable k2(F) < oo. Lemma 2. There exists a Txo= TX0(F)< oo such that for all |w| = \,k> 1, *(«,*) < TX0A2. (2.26) Moreover there exist a universal K0 < oo an*/ a k3 = k3(F) < oo íwcA that for all \u\ = l,k > k3, T(k)P{\(XX, 'For a random variable Y, o2{ Y) denotes its variance. /[ ] denotes the indicator function of the event between square brackets. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ¿/-DIMENSIONALRANDOM WALK 77 Proof. For any real random variable Y its concentration function Q(Y,') is defined by ß(y,L)=supP{y < Y< y + L}. y It is easy to see and well known (see [7, Chapters 1.1 and 2.1]) that this sup is taken on for somey, that for 0 < L, < Z^, Q(Y,L2) < (LXXL2+ I)Q(Y,LX), (2.31) and that for Yxand Y2independent, ß(T, + Y2,L)< min{Q(Yx,L),Q(Y2,L)). (2.32) If Yx, Y2,... are independent and identically distributed with distribution function G, then Theorem 3.1 of Esseen [6] gives ß|2 Y{,L] T2, = L-2 inf f This implies (2.26). The same argument shows for any L > L, P{\(Sn,tS)\ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 78 HARRY RESTEN P{|<5r_„(o>| < M(a)Ak) >l-(Sd + 8)"1 >\ (2.34) (see (2.21) for w ^ w2, and the definition of T(k) and M(u2,k) = 1 for a = u2). On the other hand, by (2.31) with L, = ¿ M(u>)Akand (2.33),the left-hand side of (2.34)is bounded above by i7^(r-ir,/2p{|<^,tó>|>¿M(wK}-1/2, so that (T- \)P{\(Xi¿>\>\M(0)Ak} < 2%2. (2.35) Now (2.32) for Yx= Y,Y2= T shows that ß(F*,L) < Q(Y,L) and this, together with (2.35),yields ß«*„ P{\XX\>TX2}<\, (2.37) and k3 = k3(F) so large that for all k > k3, \AkMM(<*,k) > r12 as well as (T(k) - iyx2x2KJ 2. Then (2.36) shows that there exists some interval [a,a + jM(co)Ak] which contains P{| This proves (2.27)for k > k3(F). To obtain (2.28) we again apply (2.33) with Y¡= <*„w> and take L = M(tS)Ak.Combined with (2.31)and (2.34)this gives 3KS(T- \yx/2M(u)Ak{( License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ¿/-DIMENSIONALRANDOM WALK 79 T(k) Í d-[( dG(y)}l( y¿/C7(y) = £{y||T| Then \d\ < c and ( y2dG<(y) > // (y, - v2)2 - // {(J, - d)2 + 2(yx - d)(y2 - d) + (y2 - d)2} dG(yx) dG(y2) \yt\ = 2f dG(y2)f (yx-d)2dG(yx). Consequently, a2{y/[|y| = f (y-d)2dG(y) + d2[ dG(y) J\y\ <ï(P{\Y\< c})~-[ y2dG>(y) + P{|y|> c) c2. J\y\<2c When this is applied to Y = (Xx,a), c = M(oi)Ak, we obtain from the above inequality: T(k)o2{(Xx,u)l[\(Xx,«>)\< M(u>)Ak]} <(26K2(P{\(Xx,œ)\ + T(k)P{|<*„W>|> M(o>)Ak}){M(u)Ak}2. Togetherwith (2.27)and(2.29) this implies(2.28). Lastly we prove (2.30). For ¿o = ¿o2(2.30) is immediate since we already observed that (2.22) holds for ¿o= ¿o2with N replaced by T. We therefore fix u =£ u2 for the remainder of the proof. Since, for any n < T, (ST,u>}is the sum of the independent random variables <5[7>I-i]n,to>and License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 80 HARRY RESTEN P{\(ST,cc}\<^2M(u>)Ak} <33Q{{ST,u),3\M(u)Ak) (by (2.31)) < 33Q{(S[Tn-,xn, < 33K5[T/n]-l'\p{\(Stà\>±M(<*)Ak})-l/2 (by(2.33)) < 33K5[T/n]-}/2{l - Q((Sn,Uy,^M(o:)Ak)y~/2 (as in (2.36)). (2.38) Now in order to prove (2.30) we only have to consider those ¿ofor which P{|<5r,|>ÍM(«K}<¿. For such an ¿o(2.38) implies for all n < T, Q{(S„,U},¡M(w)Ak) > 1 - 2X4K2nT~x. (2.39) Now apply (2.39)with n = N', where N' = N'(u,k) = min(«: P{| > p{(Su+iw - SjN.,w}>\M(a)Ak, 0 > (l - 2%2^)r/W'minP{|<5/,¿o>| > li Í I exP- 2%2 (usethe definitionof 7Y'). (2.42) Ow T O License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ¿-DIMENSIONALRANDOM WALK 81 If 2 < T/N' < 2wK¡(d + 1) + 4, we use (2.40)instead of (2.41)and obtain, as in (2.42), P{(ST,u)>2-M( > p{(Su+xw - SJN,,ti>)>\M(u)Ak, X 0 P{\(ST,co)\>^M(U)Ak} > P{|<^,co>|> M( Jn = Jkn « /[!<*>,*>!> m(k)Ak or |,*>|>4k], vr = vk = infj«: 2 (1 - J,) - iT(*) }, yr - Y? - 2 *»• (2-46) r,_, <«<»-, Yr is the sum over a bunch of ^„'s, exactly T(k) of which satisfy \(X,ax)\ < m(k)Ak and |<.V,u2>| < .4*. It is easy to see from this that the Y„ r > 1, are independent and identically distributed. Moreover, we can write Yr= 2 XJn+ 2 xn(\-jn), License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 82 HARRY RESTEN and by definition the second term contains exactly T(k) summands with 1 - J„ ¥=0. Thus if ax,a2,..., ßx,ß2,... are independent, each a¡ (/?,) with the conditional distribution of A",,given/, = 1 (respectively 7, = 0), then 2 Xn(\-Jn) (2.47) pr_i < n < r, has the same distribution as T'k) 2 ßn- (2.48) «=i Moreover, the sum (2.47) is independent of 2, <„ = ^(k)-i + iyp{JiSsl))l(l_p[Ji=:l])r^ I > 0. (2.50) Lemma 3. There exist constants TX3(F)- TXS(F)< oo such that for 2k/i < Ak < 2*/2 and any choice of the unit vector ¿o0= ¿OqE 9C*and k > 1 one has P{\Sn\< Ak for some 2k < n < 2k+x} < Tx3{Ak2-k/2}"-2 2 p 2 1ÍV 2k-l < 4M(o)k,k)Akfor0 +r,4exp - r,5A:. (2.51) Proof. Let §„ be the o-field generated by Xx,..., Xn, and consider for some fixed k the {§„) stopping times t = T(k) = min{n: 2k < n < 2k+x, \S„\ < Ak) (= oo if no such « exists), o = o(k) = inf{i'r: vr > t). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ¿-DIMENSIONAL RANDOM WALK 83 Then the set {t < oo, J, = 0 for exactly s indices / with / < t but / greater than the last vr < t} (2.52) belongs to §T. Now fix an w G % u {w3,..., tod) and set ¡M(»,k) if«E3C M*(u,k) = [1 if « G (w3,...,«,,}. Then M*(w,A:)> 1 by (2.23),and on the set (2.52), P{\(Sa,u)\>4M*(o>,k)Ak\§T} y = y(s,k) = inf{* n: 2(1-/,)-.r<*)-fj2 0 - /,) = T(k) l í-1 (note that only 0 < s < T(k) can occur). By definition y(s,k) > T(k) - s, and for each L and k > k2(F), P{\(Sy,u)\>3M*(u,k)Ak) < P{\(Snk)-s¿)\>2M*(0,k)Ak} + P{\(Sy - Sm).„W>|> M*(u,k)Ak) < (4d + 4)-'+ P (y - (T(k) -s)>L) + p[max\(Snk)-,+J - Srw_„w>|> M*(o,k)Ak} (see (2.21)and (2.24)). Now for all s < T(k), f T(k)-s + L \ P{y(s)- (T(k)-s)>L} = p\ 2 (!"//)< 7(*) - jJ Í T^+L } T(k) + L Moreover,by (2.46)and (2.27),for k > k3(F), P [J,k = 1} < P (K*,,^)! > M(uk,k)Ak for i - 1 or i = 2} <2KQT(k)-x. (2.53) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 84 HARRY RESTEN Since T(k) -* oo with k (see (2.29)) we can fix an L, independent of s < T(k), such that for all k > k3, P{y(s) - (T(k) -s)> L) < 2K0(T(k) + L)(LT(k))~l< (id + 8)"1. With L fixed in this way PÍ max\(Snk)-s+j - ST(k).s,u)\> M*(u,k)Ak\ as soon as inîuM*(u,k)Ak is large enough. Thus by (2.29) we can find k^F) < °o»such that for k > k4(F) one has on the set (2.52), P{| {T(k)-s+K¿2k ] 2 Ji>A¿2H (2.55) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ¿-DIMENSIONAL RANDOM WALK 85 (recall T(k) - s < T(k) < TX0A¡< Tx¿2kby (2.26) and Ak < 2*/2). Since T(k) - s < rI02* and P {J¡,- 1} -*0 as k -» oo (see (2.53)), standard ex- ponential estimates show that the last member of (2.55) has an upper bound of the form TX6(F)exp - Txl(F)2kuniformly in s < T(k) (see [10, Chapter 9, problems 12-16]). The first term in the last member of (2.54) is more troublesome. Recall that P is now assumed to satisfy (2.1). By taking C0 smaller if necessary we may assume that C0={zE Rd: \(z,uj> -Cj\ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 86 HARRY RESTEN and ¿02).Thus, if 8X,82,... are independent random variables, each one uniformlydistributed on C2,then P j | 2 E„ - p,| Now the corollary to Theorem 6.2 of [6] applies to the independent identically distributed (d —2) vectors {8¡)!>x whose distribution is not concentrated on any (d - 3) dimensional hyperplane and therefore the last member of (2.57) is bounded by r18(p)p-(rf-2)/2 n {4x~xAk + 1} < r19(F){^-'/2}'-2. J-3 It follows that for fixed r, p{^E[2*,(Ä:0 + r10 + 2)2*) and \(SPr,Uj)\< 4M*(uj,k)Ak for 0 < j < d] 2" +r,9(p)^-2(2*-'p{/f, = i}) ~(d-2)/2 .p\VrE[2k,(K0 + Txo+ 2)2k),'2En>2k-xP{Ex = l), { n-l \(Srr,œJ)\< 4M*(o>j,k)Ak= 4M(ù>j,k)Akfor 0 < j < 2 .(2.58) Since P{EX = 1} = a(2Xf is independent of k, the first term in the right- hand side of (2.58) is at most T20(P)exp- T2x(F)2k (see [10, Chapter 9, problems 12-16])and we obtain License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ¿-DIMENSIONAL RANDOMWALK 87 P{3v, E[2k,(K0 + r,0 + 2)2*) with |<.S>,>| < 4M*(uj,k)Akfor all 0 < j < d} < 2 i(r20 exp - T2x2k) + {T22(F){Ak2-k'2)d-2) 2*-' + P { v, E \2k, (K0 + r,0 + 2)2*) for some r < 2k-x(T(k))~l orr > (K0 + Txo+ 2)2k(T(k)yx}. (2.59) Since vr increases with r and vr > rT(k) (by definition of vr) the last term in the right-hand side of (2.59)is at most P[vn > 2* for« =[2*-,(P(A:))"1]} and we leave it to the reader to show that this term again is bounded by r23- exp - T242*.Thus, since Sv = ~2r„=xYn,the last member of (2.54) is indeed bounded by the right-hand side of (2.51) for k > max(k2,k3,k4)and suitable Tu - r,5. Obviously we can then insure the validity of (2.51) for k < max(k2,k3,k4)by increasing r,4. □ 3. Completion of the proof. Lemma 3 shows that Proposition 1 will follow once we show that there exist k5(F) < oo and K2 < oo such that for a suitable choice of wq G <3CC, 2s 2^ 2 {r>n < 4M(ujk,k)Ak for 0 < / < 2 71=1 < K2, for all k > ks, s > 1 and,4* G [2*/8,2*/2]. (3.1) Note that we can always obtain (1.4) for k < k5 by increasing Tx; moreover, (1.4) is vacuously true for A > 2k/2, and, as observed before, (1.4) follows from (2.6) when A < 2k/s. Note also that the last statement of our theorem will be immediate from Proposition 1, because (1.2)—(1.4)imply for all b > 0, P {i>(nyx\S„\ < b for some n > 2k] < 2 ^{l^l < 2'/2ty(2') for some2' < n < 2/+1} l~k < r,(p)f 2 {21/:W) +1}¿-22-^-2)/2 + exp- /r2(p) l/=* < K3Tx(F)bd~2 f00 t(t)d~2r"/2 dt + #3r,(F)2-*«,-2)/2 2k + r,(P){l - exp - r2(P)}"'exp - kT2(F)->0 (k^oo). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 88 HARRY RESTEN Moreover, it is well known that if (1.2) holds, then \p(n)~x\S*\ -» oo w.p.l or t(t)~x\B(t)\ -» oo w.p.l if and only if (1.3) holds (see [2, Theorems 5,6] or [8, §4.2.15];for S* one can also use the multidimensional analogue of Remark 2, p. 1182 in [9]). Thus to prove our theorem it suffices to prove (1.4), and this in turn has been reduced to proving (3.1). This will be done by means of certain inequalities on two dimensional concentration functions which are contained in Proposition 2 and some corollaries. We therefore begin with their proofs; the notation is as in Proposition 2. Proof of Proposition 2. By [6, formula (6.9)] sup P 2z,+ < Kj? f dBexp - \ 2 [ 0 - cos<0,M»¿/G/ (u) + (1 - cos<0,t>»]dGf (u) dGf (v) J • f [(1 - COS<0,H»+ (1 - COS<0,Ü»1 Ju,oeA,L J dGf(u) dGf(v)\ dB. (3.2) Now take C as in (1.7) and apply Holder's inequality, as in [5, p. 212] or [6, p. 294].It then followsthat the last member of (3.2)is bounded by c-'s;.,»2 Ktf d6 exp - K5C\B\' •exp^Tr-l dGf (u) dGf (v) \J\e log { dB exp - -^r (2 - cos<0,u> - cos<0,o». License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ¿-DIMENSIONAL RANDOMWALK 89 Clearly f dB exp - K5C\e\2<[ dOexp - K5C\9\2=-— . •'|0| f dBexp - -~ (2 - cos<0,u>- cos<0,t>» >| [ d»exp--^[(l-cosffi|«|) •V| + (l - cos(0,|u| cos < [ d9x exp - — [1 - cos 0,|u|] -/lö,i • sup i ¿02 exp- [ 1 - cos(è + 02|ü|sin 6 |m|C'/2 Vl ' ; |o| jsinrpIC1/2 Vl ' ' ' ' (compare [5, p. 213]) < 2 JT (l I Pi ). C 6\ |u| |sin (1.8) now follows by putting these estimates together. □ Corollary 1 is immediate from Proposition 1 by specialization (take all ft = p and A¡ = Bx X Bj). More important for our proof, though, is the more technical Corollary 2. Let ZX,Z2,... be independent identically distributed two-vec- tors and assume that there exist constants ¿, > 0, —oo < ¿2 < oo, ¿3 > 0 íucA that Px = P{Z[ (2) > ¿„ Z[ (1) < d2Z\ (2)} > 0, (3.3) as well as p2 = P{Z[ (2) > ¿„ Z\ (1) > d2Z\ (2) + ¿3 } > 0. (3.4) Thenfor all p > 0 and all n > 1, License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 90 HARRY RESTEN supP 2z,. + z < p < r■•¿{ur+ 1. (3.5) for some constant T3 = T3(d¡,p¡) < oo depending on px,p2, dx — d3 only. In fact T3 < T4(D) < oofor all p¡, d¡ and D < oo with \d2\< D, dxd3x < D, px> D~x, p2> D~x. (3.6) Moreover,for all s > 1 and p > 0,8 E#lrE[s,2s]: 2z, < p < r:teH<-+ log2). (3.7) If instead of (3.3) and (3.4) we assume that for some dx > 0, —oo < d2 < oo, ¿w¿/¿/3 > 0, p3 = P{Zx (2) > dx,Zx(1) < ¿/2Z,(2)} > 0 (3.8) as well as p4=P{Zx (2) > dx,Zx(1) > d2Zx(2) + d3 } > 0, (3.9) thenfor each s > 1, p > 0, V2 1 £•#^£[5,2*]: 2z„ < p < r5 (3.10) i (*r+i. «om>w/rA T5 = r5(^„ d¡) < oo depending only onp3,p4 and dx — d3. Moreover, T5 < T6(D) < oofor all Pi, d¡ and D < oo with \d2\< D, dxdfx < D, p3> D~\ p4 > D~x. (3.11) Lastly, assume that (3.8) and (3.9) hold and that Wx,W2,... are independent random vectors each with the distribution of2Zf^xZ¡, where A is independent of the Z¡, has the distribution given by (2.50), and that for some d4,ds, 0<¿/4< T(k)P{Jx = 1} < ¿/5 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ¿-DIMENSIONAL RANDOM WALK 91 Bx = {zE R2. z(2) > ¿„ z(l) < d2z(2)} and B2= {z E R2: z(2) > ¿„ z(l) > ¿2z(2) + ¿3 } (see figure). / z(l) = ¿2z(2) 4(1) = ¿2z(2) + ¿, z(2) = ¿, Clearly, BX,B2c (z G R2: |z| > ¿,} and [dG*(u)=Pi Moreover, for u E Bx, v E B2, \sin |o| |sin q>(u,v)\ > c = - min{¿„¿3}, (3.15) H 1 +|«2| License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 92 HARRY RESTEN and (3.5) followsfrom (1.10)with T3 = Kx(l + dxc-x)q(pxp2)-1 < tfitPrftT'O +4(1 +|¿/2|)(1+ dxdfx)}. (3.16) Recall that we took p = ¿/, so far. However, once we have (3.5) for p = ¿/, it follows for each p > 0 because any disc of radius p can be covered by at most #7(p2¿/,-2+ 1) discs of radius dx. It is immediate from (3.16) that T3 is bounded above wheneverp¡,d¡ satisfy (3.6). Also (3.7) is immediate from (3.5) because2%sr~x < 1 +log2. Next we prove (3.13) by showing that (3.3) and (3.4) hold for suitable values of px,p2 and Z replaced by W. For this purpose take Bx as above. Assume further that zx,z2,z\ E Bx satisfy z,(l) - ¿/2z,(2) < min{z2(l) - d2z2(2), z\(l) - d2z\(2)) (3.17) as well as z2(2) > z\(2). (3.18) Then z,(l) + z2(l) - z\(l) - d2(zx(2) + z2(2) - z'x(2)) < 22(0 - ¿2*2(2) < 0, (3.19) and z, (2) + z2(2) - z\ (2) > z, (2) > ¿/,. (3.20) (3.19) and (3.20) just say that zx + z2- z\ E Bx, so that zx + z2- z\ E Bx as soon as (3.17)And (3.18) hold. In agreement with the notational convention from before Lemma 1 we now take A' an independent copy of A and put wx= 2 z„ w[ = 2 z¡, w\ =wx- wx. 1-1 /-l By comparing the different orders of the quantities in (3.17) and (3.18) we now obtain P{W\ EBX}> P{A = 2,A'= 1} • f P{ZXE dzx,Z2E dz2,Z\ E dz\) Jz,,z2,z\BB, (3.17) and (3.18) hold >Ip{A = 2}P{A'=l} 6 •f P{ZxEdzx)P{Z2Edz2)P{Z'xEdz\) Jzx,z2,z\ e B, = |P{A = 2}P{A' = 1}(P{Z, E Bx })3. (3.21) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ¿-DIMENSIONAL RANDOM WALK 93 Now,by (2.50)and (3.12), P{A - 2} > Jj- (T(k)P{/, = I})2 exp - 2¿5 >\d2 exp - 2¿5. Similarly, P{A= 1} > ¿4exp-2¿5, and finally,P {Z, G Bx) = p3 (by (3.8)).Thus P{W[(2)> dx,W[(l) < d2W[(2)) = P{W[EBX)> Ksd¡p¡ exp - 4d5. (3.22) Next consider P { W\ E B2). Assume zx,z2,z\ E B2 are such that z,(l) - ¿2z,(2) > max{z2(l) - ¿2z2(2),z',(l) - d2z\(2)), (3.23) and (3.18)holds; then we get z,(l) + z2(l) - z\(l) - d2(zx(2) + z2(2) - z\(2)) > z2(l) - d2z2(2) > d3, and as before, z,(2) + z2(2) —z',(2) > ¿,. Thus, as in (3.21) and (3.22), P{W[ EB2) > P{A = 2,A'= 1} • f P{ZxEdzx,Z2E dz2,Z\ E dz\} Jzl,z2,z\SB2 (3.18) and (3.23)hold > Ksdïexp-4d5(P{Zx G52})3 >K%d¡plexp-4d5. i3-24) (3.22)and (3.24)show that (3.3)and (3.4)hold for W{instead of Z\ and px = K%d\p\exp - 4d5, p2 = Ksd¡pl exp - 4¿5. (3.25) (3.13) is therefore immediate from (3.5). Also the bound T7 < TS(D), whenever (3.11) holds together with ¿4 > D~x, d5 < D, is immediate from (3.25) and the fact that T3 < T4(D) on (3.6). Since we may everywhere in this argument replace Z, by —Z, without influencing the distribution of W[, no change is needed if Z, is replaced by —Z, in (3.8) or (3.9). Lastly, we prove (3.10). For this purpose let A,,A2,... be a sequence of independent random variables, also independent of the Z¡, and P{^ = n={l)'+\ ¡>0. (3-26) (3.26) corresponds to T(k) = 1, P{JX = 1} =\ in (2.50). Thus (3.12) is satisfied with ¿4 = d¡ = \ in this case. Set t0 = 0, and for r > \, Tr=2A,., Wr= 2 Zf (3-27) 1-1 /-T,_,+ l License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 94 HARRY RESTEN Then the Wr are independent, identically distributed and by what we just proved (3.13) holds with T7 depending only on p3,p4,dx— d3 (since now d4 = ds= 5) and T7 < r8(D) whenever (3.11) holds. Moreover, E#\rE[s,2s]: 2z(. s or t4í<2í} 2z, W-p[$At<±norit*,>ln) < K9 exp - Kxon. (3.30) Thus, the first term in the right-hand side of (3.28) is at most 2(i+l)A9exp-Ä',o!< K"' The second term in the right-hand side of (3.28) equals 2 E#\rE[Tj,Tj+x): 2z, Tj+m = 2 2*< 2z, < P, T7+I - Tj> m\ [s/2] T, + m 2 2 (P\rJ+l-Tj>m, 2 Z,Edz\ I*/2J\ •P 2z,+ < 2 2^{T, +1-T,>/«}SUPP 2^ + 2 E{rj+i-rj}TA\(j-)\l) (by(3.13)) [s/2]. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ¿-DIMENSIONAL RANDOM WALK 95 The last estimate works only for s > 2, but for í = 1 (3.10) needs no proof anyway. From (3.28), (3.31) and the estimate for the first term in the right-handside of (3.28),we finally see that (3.28)is, for s > 2, at most kxx+ t1kx2U^J + i} We now return to the proof of the bound in (3.1). The notation will be as in § 2 for the remainder of this section. We distinguish two cases: P(yt)P{y* = l} <2-5r/ (3.33) and 2-5t, < T(k)P{JÏ = 1} < 2K0. (3.34) (tj and K0 are as in Lemma 2. (3.33) and (3.34) are the only possibilities as we saw in (2.53).) For the remainder of the proof we are only interested in w components with w G %, the plane spanned by For most quantities we shall no longer indicate its dependence on k explicitly. Lemma4. There exist k6 = k6(F) < oo and KX4< oo ímcAthat the left-hand side of (3.1)is boundedby KX4forall s > 1 and all k > k6(F)for which(3.33) holds. Proof. By dropping the condition on the (o0 component and using the above notation we see that the left-hand side of (3.1) is bounded above by 2j 2^ 2 <>>,>< 4 for/ = 1,2 (3.36) Of course, n-iV_,+ l + 2 { License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 96 HARRY RESTEN By the observations made immediately following the introduction of Yr in §2 (see (2.47)) the two sums in the right-hand side of (3.37) are independent, and if we call the first sum Zr, i.e., Zr« 2 {<£0 - ./„W«! + <*„0- /, W«2}, n-c,_, + l then the two dimensional analogue of (2.32) gives supP 2,> + zy< 4,j = 1, 2 /I-l r 2KX5(rAk)-*<^-(l+\og2). r-s ak Thus we have a bound of the desired form for (3.1) as soon as Ak exceeds some strictly positive e2. We choose e, as follows: *,6 = 32*0'/2tí-,/2 + 4, Kxl = ^(\0Kx^-xf2 + 2)-\ (3.39) Kf7 1 32^ (3.40) K„ + 8K16 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ¿-DIMENSIONAL RANDOMWALK 97 (K0 and r/ are as in Lemma 2.) For the remainder of the lemma we may now assume A* < e2. (3.41) By definition of A¿ this means that there exists a unit vector Ö, G DCwith o2(Qx,k)< 2e2. By Chebyshev's inequality we then have, with p = P p{\(Sr,Qx)-p\>%ex-n-x/2} &2 = [m(ky2 cos2 then (3.44)says P {|<.SV,fi2>|> KX1{m(ky2cos\0 + sinVo)"l/2Ak) < t,/2. (3.45) At the same time Q2is a unit vector in %, and thus by (2.23),M{Jl2,k) > 1, and by (2.30), P{\(STSl2)\>\Ak) > P{\(ST,ílj)\>{M(ü2)Ak} > î,. (3.46) (3.45)and (3.46)together imply KX1[m(k)~2 cos2rf0 + sin2 sin2«p0< 4Kj7 < |, |cos (¡p0|> {. (3.47) However, we also have from (2.30): P{\(ST,ax)\>{-m(k)Ak}>r, (3.48) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 98 HARRY RESTEN and from (2.24): pp/l/ I < p{\(ST,Ux)\>^m(k)Ak and \(ST,2X)\< KX1) = P{\{ST,wx/\>\m(k)Ak and \(ST,ux/ cos < p{\(ST,a2) sin T'k) T(k) < 2i,{|<^'"2>|>^} n-1 + jz^T2o2{ < T(k)P{Jx = 1} + ^ -^ (by(2.28)andM( License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ¿-DIMENSIONAL RANDOMWALK 99 then there exists a constant/such that P{\(ST,Q3)-j\ P{|(5r,ñ4) - | = |(Sr,ß4)|< KX1) > 1 - r,/2. Apart from the change from 0, to ß4 this again is (3.44), which we showed to be impossible. Thus (3.52) must fail and no interval of length KX1can contain P{\{ST,ü3)-v\>\Kxl) >r,/4, and at least one of the inequalities P{(5r,ß3> > v + X2K„} > r,/8 (3.53) or P{(ST,ü3) must hold. For the sake of definiteness assume (3.53) holds. Since y is a median of <5r,ß3> we also have P{(5r,ß3) < v) >\. (3.55) We are almost ready to apply Corollary 2; we merely need an estimate on v, or, more generally, the distribution of \ST\. Just as in (3.51) we have for any x > 0 and k > k6(F), p{|(5>,> - p(*)p{<;?,, License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 100 HARRY RESTEN P{| P{(Sr,Q,> >{-Kxl,(ST$3) <(p + ¡Kxl)p-x(ST,üx)} > P[p- 8e,n-1/2 < (ST,QX) < p + 8e,ij"1/2, (ST,Q3) < v) >P{{ST,ü3) P{{ST,ÜX) >\Kxl,{STSt3) >{v+\Kxl)p-x{ST$x) +\KX1) > P{p- 8£,ij-l/2 < (¿r.Q,) < p + 8e,Tj-'/2, (ST,a3) > v + {-Kxl] > V8 - 11/16= ij/16. (3.61) Lastly, we observe that p{y,^5r}= pJ2^^2^j = ^{''.^7,} < P {/„ = 1 for some n < T) < ¿tj (by (3.33)), License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ¿-DIMENSIONALRANDOM WALK 101 so that with ¿i-4-*i7, d2 = {v+\Kxl)p-\ d3 = \Kxl, (3.60)and (3.61)give P{ E#\rE[s,2s]: 2{<ñ.Oi>o, + <íí1,Q3>03}<8 «-1 (because ß,,ß3 is an orthonormal basis of %) supP < x¡, i n=i / License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 102 HARRY RESTEN The distribution of W,- 2 XJn rt""»v_i+ l was given in (2.49) and (2.50). As there we take ax,a2,... independent and each with the conditional distribution of Xx given Jx = 1. Then &¡ has the conditional distribution of Xx given /, = 1 and Wr the distribution of 2^.1«,. Moreover,by (2.46), P{|< P{9(äx,u,) < ^ or License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ¿/-DIMENSIONALRANDOM WALR 103 sequel. Assume ß, = ( —sin q>x)ux+ (cos c>,)¿o2, ß2 or -ß2 = (cos ¿o0 = (m^)-2««2?), + sin2^,} • [m(k)~~}(cos P {K20M(œ0,k)@(k) <|<ä„ß2>| < i } > b2, (3.72) P{|<ä„ß2>|> K20M(Uo,k)@(k)} < 2e2. (3.73) First assume (3.71); without loss of generality we may then assume (if necessary replace ß2 by -ß2 and/or ß, by —ß,) that P{<5„ß2> >\, <ä„ß,> > 0} >i£2. (3.74) In this case we take ß3 = -J=- { -ß, + ß2), ß4 = -{=- fßi + ß2}. V2 V2 ß3 and ß4 are orthogonal unit vectors, bisecting the second (resp. first) quadrant with respect to ß„ ß2. Obviously But also, from (3.64)and (3.67),one has p{<ä„ß4> > cos ^ ,<ä„ß3> < - or the same inequality with a, replaced by —5,. Thus if we put License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 104 HARRY RESTEN ¿imï» d2=-I, ¿/3 = 2",/2, P3 = 3J, P4Œïe2» then (3.9) holds with Z,(l) = <5„ß3>,Z,(2) = <5„ß4>,and (3.8) holds with the same replacement or with —Z,(l) = <5„ß3>, —Z,(2) = <ä„ß4>. More- over, by (3.34) and (2.29) we also have (3.12) with d4 = 2~5n, ds = 2K0 as soon as k > kn for suitable k7 = k^F) < oo. Thus, by (3.63)and (3.13), supP 2 <*>;>+ *,< 4,j = 1,2 n-l P{K20M(u0,k)®(k) < <5„ß2> <\ ,<ä„ß,> > 0} >\e2 (3.76) (compare (3.74)). Since |5,| > 1 w.p.l (see (3.64)), |<5„ß2>| < | implies |<ä,,ß,>| > 2".Thus, if we define for any random vector X, X= = <*,ß,>ß, + M(o>0,k)-l{m(ky2cos2 P{<5„ß,> = <ä,,ß,> >i, P{| so that P {<£T„ß,> > cos |, or License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ¿-DIMENSIONAL RANDOMWALK 105 P { < - â„G,> > cos | ,< - ä„ß2> < 0} > ± . (3.79) If we take «1 " 21 > d2 = °> ¿3 = K2ß> P3 = 321 ' P4 = 41 C2» then (3.77) is again (3.9) with Z,(l) = <ä„ß2>,Z,(2)= <ö„ß,>; (3.78) be- comes (3.8) and (3.79) is (3.8) with Z replaced by —Z. As before we also have (3.12)with ¿4 = 2_5rjand ¿5 = 2KQfor k > k7 from (3.34)and (2.29).Thus, by (3.13)and (3.63), supP <8,/=l,2 n = \ - 4T1 { - 2 <3>i> -777 sin ç, + 2 <*>2> cos 2 (YjaJ = {M(o>0,k)©(k)yx 2 = {AkM(o>Q,k)®(k)}- r 1 r 2 <7»'wi>-r?TTC0S = AkxM(a0,kyx 2 <î>o> (3.82) «-i It follows that the condition 2 <>>,•>< 4M(w,,A;K (3.83) n=l for/ = 0 is equivalent to License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 106 HARRY KESTEN 2<^ß2> <4, n-l and condition (3.83)for/ = 1 and 2 implies 2 (YA) < 4|sin This settles the case where (3.72) holds and we are left with the case where (3.73) holds. Without loss of generality we may also assume that (3.71) fails. We begin with an estimate on the distribution of <5'7-,ß2>in this case. Analogouslyto (3.82), | p{\(ST,ü2)\>\e(k)M(UQ,k)} > r,. (3.84) Obviously, n-l n-l Moreover, for any L > 1, 2 (xnJn$2> >±Q(k)M(G>0,k)\ n-l License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ¿/-DIMENSIONALRANDOM WALR 107 (compare(2.49) and (2.50)).The right-handside of (3.85)is boundedby \eJx + LP{|<5„ß2>|> ¿e(*)JlfK*)}. Now take L = L0 (see (3.69)).Then it followsfrom (2.53),(3.70) and (3.73) that (3.85)is at most (T/L0)P{JX = 1} + L0P{|<ä„ß2>|> K2Q®M(o>0,k)} <(2/I0)Ä0 + L02e2 2<^0-/J,ß2> >±e(k)M(w0,k) >|n. (3.88) n-l Next we need an estimate for the distribution of 2 <*„(!-/„),",>, J = 1,2, n-l which is a slight variation on (3.56).E.g., take./ = 1 and write temporarily /„ = /[| 2 <*,W - TE{Xx,Ux)f> X\ n-l <(T(k)/x2)o2{(Xx,Ux)l[\(Xx,ox)\< m(k)Ak]} < K0/x2. (3.89) But also In - O - /„) = /[|<^1>|< 1»|<^,.«2>|> 1], so that 2<Í/I(l-/J,C0,>-2<^n^.> n-l n-l < 2 |<1>I>| |1 - Jn - h\ n-l <#{« License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 108 HARRY RESTEN 2 <*n(i-/j, 2 (X„(i-/„),«,>-TEix^yiiKx^)^ 1] >2x\ n-l < K0(2x~x + x~2). This inequality remains valid when 2 (Xn(l - Jn)$l) - T(k)P(k) >4x < 2K0(2x~x + x~2). (3.90) Now observethat by (2.21)and (2.24), P{|(5r,ß,)|<3} >p{|<5r,W,)|< l,\(ST,w2)\<2] > 1 - P{\(ST,»X)\> m(k)Ak) - P{|<5r,(o2>|>2^} > l-3(8¿ + 8)-'>i. Thus, for x0 as in (3.69) we have 2K0(2xöx + x0-2) < r,/4 < 1/4, and 2 (XJn&ù + T(k)ß(k)< 4x0 + 3, n-l 2<^(l-/„)^>-n^)p(A:) <4x0 n-l >p||<5r,ß,>|<3, 2 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ¿/-DIMENSIONALRANDOM WALR 109 A) ¿o [ T IP 2^n = / P< 2,<ài,Qx/+T(k)p(k)\<4x0 + 3\ /-0 l 1 <-¿ + maxP< 2 2<â,A>+T(A:)P(/c)< 4xn + 3 >i- (3.91) l Similarlywe get from (3.88)and (3.90), ¡v 2 (x„(i -JM -T(k)P(k) <4x0\ V tt + maxP< >¿0WMM), n-l Tt'(ßn$i)-T(k)p(k) < 4x0 , n-I where we used the same notation as in (2.47) and (2.48). Again there exists a 0 < /, < ¿o such that 2<Ä,,Ö2> > | Q(k)M(Uo,k), n-l T-l, 2 (ß„,ax) - T(k)P(k) < 4Xr >j. (3.92) n-l License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 110 HARRY RESTEN But, by (2.29)and (2.53)we can choose k-,(F) < oo such that for k > kn, P 2 > \ Q(k)M(o>Q,k) r-/, + i /, = P y-i2 jM(u0,k)Ak < lxP{\Xx\>(SlxyxM(u0,k)Ak\Jx = 0} < lx(P{Jx = 0}yXP{\Xx\>(8lxyxM(u0,k)Ak} < i,/8. A similar estimate gives for k > k7, P 2 (ßn$l) >4x0 <8' T-l, + l and we therefore conclude from (3.92) that 2 2,(ßn,Qx)-T(k)p(k) <8*J>-r (3.93) n-l However, with a¡, ßj and A independent, Y¡has the distribution of 2j_,/8fl + 2^xa( (see (2.46H2.50));(3.91) and (3.93)together therefore give P{\(Yx,ü2y\>i-6©(k)M(Uo,k),\(Yx,Üx)\< 12x0 + 3} >P{A = /0}P 2 (te) >±e(k)M(u0,k), n-l 2 Ä,ß, - P(A:)p(A;) i-i2 2 <«,#,>+ r(*)p(*)< 4x0 + 3 i-i > ^{A = 4>) • \ [ \ -/oP{| License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ¿/-DIMENSIONALRANDOM WALR 111 Moreover, by (2.50) and (3.34) for any /, P{A = /} > (l\)-l(TP{Jx = 1})'(1 - 2K0T~X)T > (/!)-,(2-5i,)' exp - 4/£o, as soon as K0T(k)~x < {-, i.e., for k > kn(F) if k7(F) is chosen large enough (see (2.29)).Thus, in terms of Yx,(3.94) yields for k > k-, and some K^ > 0, independent of F and k, p[vx = T+l0,\{Yx,ü2)\>i-6, |(y„ß,)|< 16(12;c0+ 3)|(y„ß2)|} > K23. We can therefore find a number xx = xx(F,k), \xx\ < 16(12x0 + 3) such that p{vx = t + /„, (y„ß2) > £, (y„ß,> < xx(y„ß2)} >\k23, (3.95) as well as p{r, = r+/0,(y„ß2)>1L, ( y„ß, ) > x, ( F„ß2)} > $ j:,,, (3.96) or both of these inequalities hold with ^{K«/0+..ß2)|<^20,|<«/0+l,ßi>|>h<«/„+l.ß2>|+i} > p{|<5/o+1,ß2>|< rnini^KnQWM^k)},^,^)^!} > P{|<5/o+„ß2>|< min{\,K20e(k)M(Uo,k)},\älo+x\> l} > 1 — e2 — 2c2 > \. Let us assume that ^{K«/0+1,ß2>|<#2» <«/0+lA> >h(«/0+A>| + 4~>Xl{%+l&l)+\}>\- (If this fails then we have p{\{%+i$2)\< *20. <«/„+!,«.) < -h<ä,0+„ß2>| -ï<^i<«/0+i,«2)-ï} >4-; License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 112 HARRY RESTEN this can be treated analogously.) Then we have from (2.50), (3.96) and (3.34), p{(y^2) >t6-k20>3l2,(yx$x) > *,(F„ß2) +{-} >P{A = l0+l] ■pU 2^+2 Ä,o2)>i, KixJ 2 * + 2 A.Qj), \ l |(«/0+i>Û2}|< K20, <«/o+i»ñi) > xx(alo+x$2) +\ , P(A = L+l} , /- \ >* P{A=l0} PiVx = T+l°' ^7,'°2) **• (y„ß,)>*,(y„ß2)} >±6K23T(k)P{Jx - l}(/0 + I)"1 > 2-^23(¿o + I)"' = J^ (3.97) say, with AT24> 0 independent of F and k. We can now complete the proof by a final application of Corollary 2 as before. (3.95) and (3.97) again are versions of (3.8) (resp. (3.9)) with Z,(l) = 2s 2^ 2^„< 12 < K.25' s> 1. (3.98) r—s i As in (3.81)-(3.83)the left-hand side of (3.98)is, however,an upper bound for the left-handside of (3.1).Thus (3.1)holds also in this last case. □ Lemmas 4 and 5 prove (3.1) with ks = max(fc6,fc7),K2 = max(Kx4,Kx9). As observed in the beginning of this section, this proves Proposition 1 and our theorem. References 1. K. L. Chung and W. H. J. Fuchs, On the distribution of values of sums of random variables, Mem.Amer. Math. Soc.No. 6 (1951).MR 12,722. 2. A. Dvoretzky and P. Erdos, Someproblems on random walk in space, Proc. Second Berkeley Sympos. Math. Statist, and Prob. (Berkeley,Calif., 1950),Univ. of California Press, Berkeley, 1951,pp. 353-367.MR 13,852. 3. H. G. Eggleston,Convexity, Cambridge Univ. Press,Cambridge, 1969. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ¿/-DIMENSIONALRANDOM WALK 113 4. K. B. Erickson, Recurrence sets of normed random walk in tLä, Ann. Probability 4 (1976), 802-828. 5. C. G. Esseen, On the Kolmogorov-Rogozin inequality for the concentration function, Z. Wahrscheinlichkeitstheorieund Verw.Gebiete 5 (1966),210-216. MR 34 #5128. 6. _, On the concentration function of a sum of independent random variables, Z. Wahrscheinlichkeitstheorieund Verw.Gebiete 9 (1968),290-308. MR 37 #6974. 7. W. Hengartner and R. Theodorescu, Concentrationfunctions, Academic Press, New York, 1973.MR 48 #9781. 8. K. Itô and H. P. McKean, Jr., Diffusionprocesses and their sample paths, Springer-Verlag, Berlinand New York; AcademicPress, New York, 1965.MR 33 #8031. 9. H. Kesten, The limit points of a normalized random walk, Ann. Math. Statist. 41 (1970), 1173-1205.MR 42 #1222. 10. J. V. Uspensky, Introductionto mathematicalprobability, McGraw-Hill, New York, 1937. Department of Mathematics, Cornell University, Ithaca, New York 14853 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use,3x-2M(>| >2-M(a)Ak) > (Sd + 8)"1}. Then, by (2.22), N'(u,k) < #(«,£) < T(k). Also, by the very definition of N', P{(SN.,u>-)>\M(U)Ak] > (l6)-\d+ I)"1 (2.40) or the inequality holds with (SN.,u} replaced by —(SN.,cS). For the sake of definiteness assume (2.40). Then any interval containing {SN.,cd) with a probability larger than (16¿/+ 15)(16¿/+ 16)-1 must contain points to the right of ^ M(u)Ak. By (2.39) there exists such an interval of length \ M(u)Ak which contains j,>| > 4M*(u,k)Ak for some 0 < j < d) d < P{t < oo} 2 P{\(Sa,u¡)\> 4M*(u,k)Ak\T < oo} <|P{t < oo}. j-o After taking complements with respect to {t < oo} and using the fact that o equals some vr, we obtain P{\S„\< Ak for some 2k < « <2k+x) = P{t < oo} < 2P{t < oo,\(Sg,Uj)\< 4M*(uj,k)Akfor all 0 < j < d) < 2P [3vr E [2k, (K0 + r,0 + 2)2*) with |2>" 2 £{<^,